## Archive for December, 2008

### Mathematical models for tuberculosis

December 30, 2008

In previous posts I made some remarks on mathematical models for diseases and/or the immune system. I also had a post about tuberculosis. Now I came across the web page of Denise Kirschner where there are a lot of links to her publications on modelling TB, the immune system, HIV and related topics. You can also see a related video of a talk of hers (from June 19, 2007) on the web site http://videocast.nih.gov.

In a paper of Wigginton and Kirschner (Journal of Immunology 66, 1951) the authors introduce a mathematical model to describe the interactions of the immune system and the TB bacterium within the lung. This is a system of twelve ODE. The unknowns are two populations of bacteria (inside or outside macrophages), three populations of macrophages, three populations of T cells (Th0, Th1 and Th2) and the concentrations of four cytokines (interferon $\gamma$, IL-4, IL-10 and IL-12.) A lot of detail has been included concerning models for the interaction between the different players and in extracting values from the literature for the many parameters which occur. The goal is to understand the different outcomes of disease: acute infection, latent infection and reactivation.

The ODE are solved on the computer. As far as I could see there has been no general mathematical analysis of the properties of solutions of this system done. It may just be too big and complicated but I would be interested to see if something could be done in that direction. The numerical results apparently show convergence to a stationary state and convergence to a limit cycle in different situations. This model has been further extended by Kirschner and collaborators in other papers. In one paper a model with two compartments is introduced (lung and lymph node) where dendritic cells are also included. Another paper includes CD8+ T cells and TNF$\alpha$. What I like about this work is that it seems to be making real contact between mathematical modelling and the details of immunology, going beyond the simplest model systems.

### HIV controllers

December 20, 2008

On 12.12 I heard a talk by Bruce Walker of Partners AIDS Research Centre, Massachusetts General Hospital, with the title ‘Prospects for an AIDS vaccine’. He started by saying that, given the difficulties which have been experienced in trying to develop a vaccine which could actually prevent or cure AIDS, another strategy which should be considered is to develop a vaccine which would bring people infected with HIV into a state where they could maintain a constant low viral load (and thus a symptom-free state) without medication. A fact which suggests that this might be possible is the existence of ‘HIV controllers’, people who can do this spontaneously. That is to say, although infected with HIV they can maintain very low viral loads for very long times (perhaps indefinitely) without needing drugs. Within the class of controllers there is a special type, the ‘elite controllers’ where the virus load is undetectable with available techniques. If it was known what is special about controllers or elite controllers this might provide the basis for developing the desired vaccine. There is a project to do tests on these controllers to try to find out what is going on. More information can be found on the web page http://www.hivcontrollers.org/.

What has been discovered so far? One of the problems in fighting HIV is that the virus evolves very fast to produce different quasispecies which can evade any tool being used to combat the virus. It appears that those variants of the virus found in controllers are different from those found in the general population of individuals infected with HIV. The viruses in controllers are less fit. Moreover it seems that this is not due to these people having been infected by less fit viruses – rather they have evolved to become less fit in these patients. When HIV evolves to escape attacks by the immune system or drugs it is moving in a vast genetic landscape. It seems that in controllers the direction of this evolution is biased and it ends up in regions of lower
fitness. But what is it that creates this bias? There has been a large scale genetic analysis of SNPs (single nucleotide polymorphisms) in contollers to look for special genetic characters. This gave a strong indication that a region of the genome which is involved is the HLA (human lymphocyte antigen) region. This suggests a direct involvement of the immune system and in particular of the MHC molecules (major histocompatibility complex) which are coded for by HLA genes. There are also signs that cytotoxic T-cells (CD8+) have an important role to play. The project continues. It represents a hope that an AIDS vaccine can finally be developed and, perhaps, a useful new strategy for developing vaccines for other ‘difficult’ diseases.

### Shock waves, part 2

December 14, 2008

In a previous post I mentioned the recent work of Demetrios Christodoulou on shock waves. On 08.12 I heard a talk by Christodoulou on this subject and since then he has explained some of the most important points of this work to me in more detail. Here I will present a little of what I learned. These results concern solutions of the relativistic Euler equations in Minkowski space. According to Christodoulou analogous results could be obtained in the non-relativistic case but no details of this have been published. The initial data is given on a hyperplane and is assumed to coincide outside a compact set with a constant state where the fluid is at rest with constant density. On the compact set the data is close to the constant state in a suitable sense.

The object of study is the maximal smooth solution evolving from the given data and its future boundary. This boundary will be non-empty exactly in the case when a shock is formed. Necessary and sufficient conditions are given for there to be a shock. Most of the results concern a fluid which is isentropic and irrotational. The two conditions are intimately connected and cannot be assumed independently of each other. These results do have consequences for the general case since a sufficiently large region exists where the extra conditions are satisfied. Here I will concentrate on the isentropic and irrotational case. A central point is that while the solution becomes singular at the future boundary it is actually smooth up to and including the boundary with respect to a non-standard differential structure. Key computations are done in coordinates which define this different kind of smoothness. One of these coordinates is constant on sound cones of the solution being considered. The condions for shock formation depend very much on the sign of the quantity $\rho f''(\rho)+2f'(\rho)$ at the constant state, where $p=f(\rho)$ is the equation of state of the fluid. This is the same as the sign of the quantity $H$ introduced in the book. If this quantity actually vanishes on the constant state then there are no shocks. For an irrotational and isotropic flow the evolution equations of the fluid can be written as a quasilinear wave equation for a scalar function $\phi$. In the case that $H$ is identically zero for a given equation of state this coincides with a geometric equation which is obtained as follows. Suppose that a timelike hypersurface in five-dimensional Minkowski space can be expressed as the graph of a function $\phi$ on ${\bf R}^4$. Suppose in other words that the hypersurface is of the form $(x_0,x_1,x_2,x_3,\phi(x_0,x_1,x_2,x_3))$. Then the condition that this hypersurface has vanishing mean curvature is equivalent to the equation of motion for this particular type of fluid. The fluid is related to the Chaplygin gas which has been studied in cosmology in recent years. It has equation of state $p=-A\rho^{-1}$ for a constant $A$ and satisfies $H=0$. This type of fluid originally came up in aerodynamics in the early years of the twentieth century. Sergei Chaplygin, after whom this fluid is named, seems to have been quite a prominent figure since the town he grew up in is now named after him, as is a crater on the moon. For fluids under normal physical conditions $\rho f''(\rho)+2f'(\rho)>0$ but, as pointed out in the book, there are physical situations where the opposite sign occurs.

### Lyapunov-Schmidt reduction

December 7, 2008

At an early age we learn how to tackle the problem of solving $n$ linear equations for $n$ unknowns. What about solving $n$ nonlinear equations for $n$ unknowns? In general there is not much which can be said. One promising strategy is to start with a problem whose solution is known and perturb it. Consider an equation $F(x,\lambda)=0$ where $x\in {\bf R}^n$ should be thought of as the unknown and $\lambda\in {\bf R}$ as a parameter. The mapping defined by $F$ is assumed smooth. Suppose that $F(0,0)=0$ so that we have a solution for $\lambda=0$. It is helpful to consider the derivative $N=D_x F(0,0)$ of $F$ with respect to $x$ at the origin. If the linear map $N$ is invertible then we are in the situation of the implicit function theorem. The theorem says that there exists a smooth mapping $g$ from a neighbourhood of the zero in ${\bf R}$ to ${\bf R}^n$ such that $F(g(\lambda),\lambda)=0$. It is also (locally) unique. In other words the system of $n$ equations has a unique solution for any parameter value near zero.

What happens if $N$ is degenerate? This is where Lyapunov-Schmidt reduction comes in. Suppose for definiteness that the rank of $N$ is $n-1$. Thus the kernel $L$ of $N$ is one-dimensional. We can do linear transformations independently in the copies of ${\bf R}^n$ in the domain and range so as to simplify things. Let $e_1,\ldots,e_n$ be the standard basis in a particular coordinate system. It can be arranged that $L$ is the span of $e_1$ and the range of $N$ is the span of $e_2,\ldots,e_n$. Now define a mapping from ${\bf R}^{n-1}\times {\bf R}^2$ to ${\bf R}^{n-1}$ by $G(x_2,\ldots,x_n,x_1,\lambda)=P(F(x_1,\ldots,x_n,\lambda))$ where $P$ is the projection onto the range of $N$ along the space spanned by $e_1$. Things have now been set up so that the implicit function theorem can be applied to $G$. It follows that there is a smooth mapping $h$ such that $G(h(x_1,\lambda),x_1,\lambda)=0$. In other words $(x_1,x_2,\ldots,x_n)$ satisfy $n-1$ of the $n$ equations. It only remains to solve one equation which is given by $H(x_1,\lambda)=F^1(x_1,h(x_1,\lambda),\lambda)=0$. The advantage of this is that the dimensionality of the problem to be solved has been reduced drastically. The disadvantage is that the mapping $h$ is not known – we only know that it exists. At first sight it may be asked how this could possibly be useful. One way of going further is to use the fact that information about derivatives of $F$ at the origin can be used to give corresponding information on derivatives of $H$ at the origin. Under some circumstances this may be enough to show that the problem is equivalent to a simpler problem after a suitable diffeomorphism, giving qualitative information on the solution set. The last type of conclusion belongs to the field known as singularity theory.

My main source of information for the above account was the first chapter of the book ‘Singularities and groups in bifurcation theory’ by M. Golubitsky and D. Schaeffer. I did reformulate things to correspond to my own ideas of simplicity. In that book there is also a lot of more advanced material on Lyapunov-Schmidt reduction. In particular the space ${\bf R}^n$ may be replaced by an infinite-dimensional Banach space in some applications. An example of this discussed in Chapter 8 of the book. This is the Hopf bifurcation which describes a way in which periodic solutions of a system of ODE can arise from a stationary solution as a parameter is varied. This is then applied in the Case study 2 immediately following that chapter to study the space-clamped Hodgkin-Huxley system mentioned in a previous post.